Probability that the system works I have a system. That system has 4 components, and it can only works if 2 of the 4 components works correctly. 
If every single component, independently,works correctly with a probability of 0,9. What is the probability that the system will work?
 A: Hint: The probability that it works correctly is equal to $1$ minus the probability that it doesn't work correctly.
The only cases where it doesn't work correctly are (I) exactly 0 of the 4 components work correctly, and (II) exactly 1 of the 4 components work correctly. So compute $P(\text{Case I})$ and $P(\text{Case II})$; the answer is
$$
1 - P(\text{Case I}) - P(\text{Case II})
$$
To deal with Case II, note that there are four ways that exactly one of them can work - the first one can work, or the second one can work, or the third, or the fourth.
A: The probability that all the components are bad is $(0.1)^4$. For the probability that exactly $3$ are bad, call the components  A, B, C, and D. The probability A, B, C are bad and D is good is $(0.1)^3(0.9)$. The same is true for the other $3$ possibilities of $3$ bad and $1$ good. So the probability exactly $3$ components are bad is $(4)(0.1)^3(0.9)$.
It follows that the system is down with probability $(0.1)^4+(4)(0.1)^3(0.9)$. The probability the system is working is therefore
$$1-(0.1)^4-(4)(0.1)^3(0.9).$$
